Hyers–Ulam stability of bijective $\varepsilon$-isometries between Hausdorff metric spaces of compact convex subsets

Abstract

Let X (resp. Y) be a real Banach space such that the set of all $w^*$-exposed points of the closed unit ball $B(X^*)$ (resp. $B(Y^*)$) is $w^*$-dense in the unit sphere $S(X^*)$ (resp. $S(Y^*)$), (cc(X), H) (resp. (cc(Y), H)) be the metric space of all nonempty compact convex subsets of X (resp. Y) endowed with the Hausdorff distance H, and $f:(cc(X),H)\rightarrow (cc(Y),H)$ be a standard bijective $\varepsilon$-isometry. Then there is a standard surjective isometry $g:cc(X)\rightarrow cc(Y)$ satisfying that $(1)\, g|_{X}$ (the restriction of g on $\{\{u\}, u\in X\}$) is a surjective linear isometry from $\{\{u\},u\in X\}$ onto $\{\{v\},v\in Y\}$ and $g(A)=\cup _{a\in A}g|_{X}(\{a\})$ for any $A\in cc(X)$; $(2)\, H(f(A),g(A))\le 3\varepsilon$ for any $A\in cc(X)$.